In-situ characterization of grain nucleation and coarsening in metallic microstructures using synchrotron radiation

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and coarsening in metallic microstructures

using synchrotron radiation

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Science and Engineering, of the Delft University of Technology, Delft, The Nether-lands.

This research is supported by the Dutch Technology Foundation STW, which is a part of the Netherlands Organization for Scientific Research (NWO) and partly funded by the Ministry of Economic Affairs, Agriculture and Innovation (07949).

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and coarsening in metallic microstructures

using synchrotron radiation

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universtiteit Delft,

op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 23 oktober 2012 om 15:00 uur

door

Hemant SHARMA

Master of Science in Materials Science and Engineering, Delft University of Technology

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Prof. dr. ir. J. Sietsma

Copromotor: Dr. ir. S.E. Offerman

Samenstelling promotiecommissie: Rector Magnificus,

Prof. dr. ir. J. Sietsma, Dr. ir. S.E. Offerman, Prof. dr. B.J. Thijsse, Prof. dr. ir. L.A.I. Kestens, Prof. dr. D. Juul Jensen, Prof. dr. A. Godfrey, Dr. J. Wright,

Prof. dr. I.M. Richardson,

voorzitter

Technische Universiteit Delft, promotor Technische Universiteit Delft, copromotor Technische Universiteit Delft

Technische Universiteit Delft Technical University of Denmark Tsinghua University Beijing

European Synchrotron Radiation Facility Technische Universiteit Delft, reservelid

c

2012, Hemant Sharma

ISBN 978-94-6182-177-5

Keywords: 3DXRD, Synchrotron Radiation, Indexing, Peak Overlap, Phase Trans-formation, Grain Nucleation, Grain Coarsening, Iron Alloys

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2 A multi−purpose furnace for in−situ studies of polycrystalline mater-ials using synchrotron radiation 11 2.1 Introduction . . . 12 2.2 Design Requirements . . . 15 2.3 Furnace design . . . 16 2.3.1 Furnace body . . . 16 2.3.2 Specimen chamber . . . 19 2.3.2.1 Atmosphere control . . . 20

2.3.3 Heating element and specimen holder . . . 20

2.3.4 Furnace control . . . 21

2.4 Furnace characteristics and performance . . . 22

2.5 Experiments . . . 22

2.6 Future developments . . . 26

2.7 Conclusions . . . 26

3 Pre−processing of 3DXRD diffraction data 33 3.1 Introduction . . . 34

3.2 3DXRD technique . . . 35

3.3 Characterization of peaks . . . 37

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3.3.2 Peak fitting . . . 40

3.3.2.1 Minimization . . . 41

3.3.2.2 Implementation and workflow . . . 42

3.3.3 Merging inω . . . . 45

3.3.4 Limitations . . . 46

3.4 Fitting global parameters of the experimental setup . . . 48

3.4.1 Parameters affecting the position of the diffraction spots on the detector . . . 49

3.4.2 Parameters affecting the position of the diffraction spots inω 51 3.4.3 Implementation . . . 53

3.5 Test cases . . . 55

3.5.1 Peak searching and peak fitting . . . 55

3.5.1.1 Simulated data . . . 55

3.5.1.2 3DXRD experiment . . . 56

3.5.2 Global parameters− simulated data . . . 60

3.6 Summary . . . 62

4 Determination of volume, centre of mass, crystallographic orienta-tion and strain state of grains 69 4.1 Indexing using a surface scanning technique . . . 71

4.1.1 Introduction . . . 71

4.1.2 3DXRD technique: setup and principles . . . 72

4.1.2.1 Grain volume . . . 74

4.1.2.2 Grain orientation and position . . . 76

4.1.3 Indexing . . . 78 4.1.3.1 Classical approaches . . . 78 4.1.3.2 Present approach . . . 80 4.1.3.3 Implementation . . . 84 4.1.3.4 Limitations . . . 91 4.1.4 Refinement . . . 92 4.1.5 Strains . . . 93 4.1.6 Test cases . . . 94 4.1.6.1 Un−strained simulations . . . 95 4.1.6.2 Strained simulations . . . 102

4.2 Indexing using Friedel pairs . . . 106

4.2.1 Friedel pairs . . . 107

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4.2.3 Indexing using asymmetric Friedel pairs . . . 112 4.2.4 Results . . . 114 4.2.4.1 Simulations . . . 114 4.2.4.2 3DXRD Experiment . . . 116 4.3 Summary . . . 118 4.4 Bibliography . . . 119

5 Observation of changing crystal orientations during grain coarsening 125 5.1 Introduction . . . 126

5.2 Experimental details . . . 127

5.2.1 Specimen . . . 127

5.2.2 3DXRD experiment . . . 128

5.3 Data analysis method . . . 132

5.3.1 Grain volume . . . 132

5.3.2 Mosaicity and rotation of the average plane normal of grains 133 5.4 Results . . . 133

5.5 Discussion . . . 136

6 The role of orientation relationships during nucleation of austenite 151 6.1 Introduction . . . 152

6.2 Experimental methods . . . 155

6.2.1 Material . . . 155

6.2.2 3DXRD Experiment . . . 155

6.3 Data analysis method . . . 158

6.3.1 Pre−processing . . . 158

6.3.2 Fraction transformed . . . 159

6.3.3 Indexing and refinement . . . 159

6.3.4 γ − α neighbour detection . . . 162

6.3.5 Orientation Relationships . . . 163

6.4 Results and discussion . . . 163

7 Grain coarsening in Niobium containing steels studied by 3DXRD 189 7.1 Introduction . . . 190

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7.2.1 Materials . . . 191

7.2.2 3DXRD Experiments . . . 191

7.2.3 Data analysis method . . . 194

7.3 Results and discussion . . . 195

7.4 Conclusions . . . 209 7.5 Bibliography . . . 209 Summary 213 Samenvatting 219 List of Figures 225 List of Tables 229 Acknowledgements 231 List of Publications 233 Curriculum Vitæ 235

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I

NTRODUCTION

Most metals in use are polycrystals: aggregates of multiple crystals (grains), each with a different crystallographic orientation. The size distribution, spatial distribu-tion and crystallographic orientadistribu-tion distribudistribu-tion of these grains have a direct in-fluence on the mechanical properties of the material. For example, the Hall−Petch relationship shows that the strength of a material is directly proportional to the inverse of the square root of the average grain size[1, 2]. During manufacture, the evolution of the microstructure is controlled mainly by: (i) grain nucleation and grain growth during phase transformations; (ii) grain coarsening at high temperat-ures and; (iii) deformation characteristics including recrystallization. Even though numerous efforts have been made, the understanding of these three processes in metals is not yet complete. A major contributing factor to this has been the lack of suitable experimental techniques, which allow for in−situ studies.

The advancement of 3rd_{generation synchrotron sources has opened up vast} opportunities in materials research, especially due to the non−destructive nature of the techniques. In particular for polycrystals, a number of techniques, such as X−Ray Phase−Contrast Tomography (PCT) [3], Differential Aperture X−Ray Micro-scopy (DAXM)[4], Three−Dimensional X−Ray Diffraction (3DXRD) microscopy [5] and Diffraction Contrast Tomography (DCT)[6, 7] have been used. 3DXRD is very promising for in−situ studies of polycrystalline materials during thermo−mechani-cal processing due to the unique combination of material properties that can be measured, namely: position, crystallographic orientation, volume and strain

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state of the individual grains, which can be determined in a non−destructive manner. However, in order to extend the 3DXRD technique from its present state of a few demonstrative examples to being applied for tracking of

statist-ically−relevant number of grains with a good time resolution during solid−state phase−transformations, at least two major shortcomings need to be overcome: (i) the limitations posed by the setup of the 3DXRD technique on the design of suitable furnaces for carrying out controlled heat treatments on the materials and; (ii) the limitations on the reduction/ analysis of data acquired from specimen consisting of thousands of grains using 3DXRD.

At high temperatures, grains in polycrystalline materials grow in order to re-duce the total interface energy of the system. According to the general theory of grain coarsening[8], the average grain size during isothermal annealing should be directly proportional to the square root of the annealing time. However, this relationship is very rarely observed, even for ultra−pure metals, with most exper-iments showing a faster decay in the rate of grain coarsening (the exponent for time is smaller than 0.5). This effect is commonly attributed to solute drag (slowing down of grain boundaries by solute atoms present in the matrix), to non−regular microstructures or to the presence of texture[8]. However, even after incorporating all the aforementioned factors, models for accurately predicting the rate of grain growth are non−existent.

As compared to the number of experiments carried out on grain coarsening, experimental observations of the process of solid−state nucleation are far too few, the main limitation being the difficulty in obtaining information about all the parameters affecting the nucleation rate. In particular, the activation energy for nucleation, which determines the barrier to nucleate, is very difficult to determine. The first efforts were calculations by Clemm and Fisher, who showed that the effectiveness of the potential nucleation sites is highest for grain corners, followed, in order, by nucleation at grain edges, faces and intra−granular nucleation [9]. A study by Lange et al. [10] on the nucleation rate of ferrite (α) in austenite (γ) revealed that the activation energy for nucleation was much smaller than that predicted by the models of Clemm and Fisher[9], which Lange et al. [10] reasoned by using low energies of theα−γ interfaces, which are created during nucleation.

In a recent study using synchrotron radiation, Offerman et al. [11] concluded that the activation energy for the nucleation ofα in γ was even lower by two orders of magnitude than that calculated by Lange et al. [10]. Similarly, Savran et al. [12] showed the same to be the case during nucleation of γ in α during heating.

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However, in both the studies of Offerman et al.[11] and Savran et al. [12], an overall activation energy was calculated because information about the position of the nuclei in the matrix and the crystallographic orientation of the grains could not be obtained. In reality, the total nucleation rate is the sum of the nucleation rates at different types of potential nucleation sites.

1.1 Technological relevance

Studying nucleation, growth and coarsening is of importance not only during the production of steels (to control the final microstructure), but also during the use of steels in high temperature applications such as fire−resistant steels or steels for energy conversion systems. The conventional method to improve the strength of steels for use at high temperatures (and to reduce the rate of grain coarsening at high temperatures during production) is to introduce precipitates with high number densities, which hinder the movement of grain boundaries and of dislo-cations during mechanical deformation. However, precipitates have two obvious disadvantages: (i) due to the increased rates of coarsening as temperatures are increased, the number density, and thus the effectiveness in pinning the grain boundaries, of the precipitates decreases and; (ii) easier dislocation climb and glide at higher temperatures reduces the effectiveness of precipitates themselves. This thesis investigates the changes occurring at the level of individual grains during heating and during annealing at high temperatures, which can then be used to design steels with optimized properties for application at high temperatures and control the final microstructure during production. The results presented in this thesis, albeit of a fundamental nature, have direct relevance for the design and the production steels for applications at high temperatures.

1.2 Aim of the thesis

The aim of the thesis is two−fold:

(i) To extend the application of synchrotron techniques, specifically 3DXRD microscopy, to study microstructural processes in metallic alloys. This includes overcoming the limitations of the technique, such as difficulties in carrying out accurate heat treatments, limited number of grains that can be observed, limita-tions due to the data analysis etc. However, instrumentation related issues, such as detector resolution, response time etc., are not investigated.

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(ii) To apply 3DXRD to enhance our understanding of two fundamental processes in metals: grain nucleation and grain coarsening.

1.3 Contents of the thesis

The thesis is divided into two parts: Chapters 2, 3 and 4 dealing with the improve-ments made to the 3DXRD technique and; Chapters 5, 6 and 7 describing the added insight obtained on the fundamental processes of grain nucleation and coarsening in model iron−based alloys.

Chapter 2 details a special furnace developed for the 3DXRD type of experiments. The furnace conforms to all the requirements of the synchrotron radiation tech-nique as well as the qualities required for conducting research on polycrystalline materials during heat treatments.

Chapters 3 and 4 describe a software framework for the pre−processing of 3DXRD data (Chapter 3) and the techniques to obtain grain−specific information about volume, centre−of−mass position, crystallographic orientation and strain state (Chapter 4) simultaneously for thousands of grains. The software techniques de-veloped are faster when handling more grains than other technique for the analysis of 3DXRD data.

Chapter 5 describes unique experiments on the coarsening ofγ grains in a model Fe−2 wt% Mn alloy at 1000◦_{C. By combining information about the volume,} aver-age crystallographic orientation and orientation spread of individual grains, a new mechanism responsible for the often observed reduction in the rate of coarsening is proposed. Furthermore, multiple modes of grain coarsening are presented.

Results of the effect of crystallographic orientation relationships on the nucle-ation of theγ−phase in a model Fe−C−Mn−Ti alloy during heating are described in Chapter 6. For the first time, theΨ−parameter, which represents the effects of the shape of the nucleus and the interfacial energies, is calculated for five dif-ferent types ofγ−nuclei depending on the number of α neighbours with special orientation relationships.

Grain coarsening in two specially designed alloys with high Nb:C ratios for high temperature applications is studied in Chapter 7. Grain coarsening in these alloys is in complete contrast with ‘normal’ grain coarsening in polycrystalline materials, such as the results shown in Chapter 5.

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1.4 Bibliography

[1] E. O. Hall, “The deformation and ageing of mild steel: III Discussion of results,”

Proceedings of the Physical Society. Section B, vol. 64, no. 9, pp. 747–753, 1951.

[2] N. J. Petch, “The cleavage strength of polycrystals,” Journal of the Iron and

Steel Institute, vol. 174, pp. 25–28, 1953.

[3] J. Baruchel, P. Bleuet, A. Bravin, P. Coan, E. Lima, A. Madsen, W. Ludwig, P. Pernot, and J. Susini, “Advances in synchrotron hard X-ray based imaging,”

Comptes Rendus Physique, vol. 9, no. 5-6, pp. 624–641, 2008.

[4] B. C. Larson, W. Yang, G. E. Ice, J. D. Budai, and J. Z. Tischler, “Three-dimensional X-ray structural microscopy with submicrometre resolution,”

Nature, vol. 415, no. 6874, pp. 887–890, 2002.

[5] H. F. Poulsen, Three-Dimensional X-Ray Diffraction Microscopy- Mapping

Polycrystals and Their Dynamics. Berlin: Springer, 2004.

[6] W. Ludwig, S. Schmidt, E. M. Lauridsen, and H. F. Poulsen, “X-ray diffraction contrast tomography: A novel technique for three-dimensional grain mapping of polycrystals. I. Direct beam case,” Journal of Applied Crystallography, vol. 41, no. 2, pp. 302–309, 2008.

[7] G. Johnson, A. King, M. G. Honnicke, J. Marrow, and W. Ludwig, “X-ray dif-fraction contrast tomography: A novel technique for three-dimensional grain mapping of polycrystals. II. The combined case,” Journal of Applied

Crystallo-graphy, vol. 41, pp. 310–318, 2008.

[8] F. J. Humphreys and M. Hatherly, Recrystallization and related annealing

phenomena. Oxford: Elsevier, 2004.

[9] P. J. Clemm and J. C. Fisher, “The influence of grain boundaries on the nucle-ation of secondary phases,” Acta Metallurgica, vol. 3, no. 1, pp. 70–73, 1955. [10] W. F. Lange, M. Enomoto, and H. I. Aaronson, “The kinetics of ferrite nucleation

at austenite grainboundaries in FeC alloys,” Metallurgical Transactions A

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[11] S. E. Offerman, N. H. van Dijk, J. Sietsma, S. Grigull, E. M. Lauridsen, L. Mar-gulies, H. F. Poulsen, M. T. Rekveldt, and S. van der Zwaag, “Grain nucle-ation and growth during phase transformnucle-ations,” Science, vol. 298, no. 5595, pp. 1003–1005, 2002.

[12] V. I. Savran, S. E. Offerman, and J. Sietsma, “Austenite nucleation and growth observed on the level of individual grains by Three-Dimensional X-Ray Dif-fraction microscopy,” Metallurgical and Materials Transactions A - Physical

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A

MULTI

−

PURPOSE FURNACE FOR

IN

−

SITU STUDIES OF

POLYCRYSTALLINE MATERIALS USING

SYNCHROTRON RADIATION

Abstract

A multi−purpose furnace designed for studies using synchrotron radiation on poly-crystalline materials, namely metals, ceramics and (semi−) crystalline polymers, is presented. The furnace has been designed to carry out Three−Dimensional X−Ray Diffraction (3DXRD) measurements, but can also be used for other types of synchrotron radiation research. The furnace has a very low thermal gradient across the specimen (< 0.2◦_{C/mm). A thermocouple welded to the specimen is} used to accurately determine the temperature. The furnace can be rotated over an angle of 90◦_{in order to determine the crystallographic orientation and the} posi-tion of each individual grain. It is possible to follow the growth kinetics of all the grains in the illuminated volume of the specimen. The specimen environment can be controlled varying from vacuum (up to 10−5_{mbar) to gas or air filled. The} maximum temperature of operation is 1500◦_{C, with the possibility of achieving} high heating (up to 20◦_C_{/s) and cooling rates (up to 30}◦_C_{/s without quenching}

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synchrotron radiation

gas). 3D maps of the microstructure of the specimen can be generated at elevated temperatures by positioning a high−resolution detector close to the specimen. An example of a simulation of the heat affected zone during the thermal cycle of a weld in a TRIP steel carried out using the furnace is shown. More examples are shown in Chapters 5, 6 and 7. The unique characteristics of the furnace presented here open the possibility of new fields of research in materials science using synchrotron radiation.

2.1 Introduction

In recent years, high−energy X−ray radiation, available at 3rd_generation synchro-tron sources such as the European Synchrosynchro-tron Radiation Facility (ESRF) in Gren-oble (France), the Advanced Photon Source (APS) in Argonne (USA), and Spring−8 in Nishi Harima (Japan), has become more and more important for studies in ma-terials science. In particular for polycrystalline mama-terials, X−ray Phase−Contrast Tomography (PCT)[1, 2], Differential Aperture X−ray Microscopy (DAXM) [3, 4], 3−Dimensional X−ray Diffraction Microscopy (3DXRD) [5, 6, 7, 8], and Diffraction Contrast Tomography (DCT)[9, 10, 11] have been used. PCT is a powerful tech-nique to study material systems with an inhomogeneous density distribution, e.g., silicon particles can be visualized in cast aluminium alloys[2], but the crystal ori-entations cannot be determined. DAXM has nano−metre resolution, but is limited by small specimen volumes and long acquisition times. The 3DXRD technique, developed jointly by scientists at beamline ID11, ESRF and RISØ, Denmark in the 1990s, has shown to be a powerful tool for carrying out in−situ studies in metals. The DCT technique can be considered a variant of 3DXRD.

The 3DXRD technique allows for non−destructive characterization of grains in terms of volume[7, 12], orientations [13, 14] and stresses [14, 15, 16] in bulk microstructures. The experimental setup is rather similar to X−ray tomography, the difference being that, in tomography one probes the attenuation of the direct beam and reconstructs the density of the specimen. However, in 3DXRD, the diffracted signal from different crystalline grains is used to reconstruct the microstructure within the material. In this way, a 3D image of the microstructure can be obtained, even for materials without a density difference, with micrometre resolution. In comparison, Focused Ion Beam (FIB) 3D Electron Back−Scatter Diffraction (EBSD) [17] has a better spatial resolution, but, being a surface technique, is destructive and ex−situ in nature and the volumes that can be investigated are limited. In general, 3DXRD can be used in three modes. In the ‘fast mode’, the characteristics of

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each grain (for example volume, position of centre−of−mass and crystallographic orientation) can be obtained with a time resolution of the order of a few minutes. On the other hand, in the ‘slow mode’, full 3D mapping of the grain position, shape and orientation can be carried out, with a mapping precision of 5×5×1 µm3_{, but} with a worse time resolution of the order of a few hours. In the ‘high−resolution mode’, deformation structures of the material can be investigated by placing the detector at a large distance from the specimen[15]. The 3DXRD setup is shown in Figure 2.1. In order to get all the grains under diffraction a number of times, the specimen is rotated around the vertical axis (ω).

The present chapter describes a furnace specially designed to carry out 3DXRD measurements at temperatures up to 1500◦_{C, while meeting all the requirements} of the setup of the 3DXRD technique. The furnace is suited for studies of poly-crystalline materials including metals, ceramics and semi−crystalline polymers. Although the furnace was designed for 3DXRD experiments, it can as well be used for X−ray tomography and DCT in the low temperature mode (see §2.6 for de-tails). In addition, the furnace can also be used for other synchrotron radiation experiments such as High Resolution Powder Diffraction and Small Angle X−ray Scattering (SAXS). The furnace overcomes various shortcomings of the other fur-naces currently available for 3DXRD experiments. For example, while operating in the high temperature range, the specimen can be rotated over an angular range of 90◦_{, as compared to a previous furnace reported by Margulies et al.} _{[18], in} which rotation is limited to an angle of 9◦_{only, which is not enough to bring all the} grains under diffraction sufficient number of times in order to determine the crys-tallographic orientation of all the grains. Also, the furnace of Margulies et al.[18] cannot be used with high−resolution detectors due to the geometrical constraints arising from the design of such detectors. In comparison, the furnace reported by Bellet et al.[19] has the advantage that, being a tomography furnace, it can be rotated over an angle of 360◦_{. However, it suffers from the disadvantage that the} maximum temperature that can be reached is 1030◦_{C and, because the specimen} is heated directionally from the bottom, has a considerable thermal gradient across the specimen. Furthermore, the high temperature stability of the furnace of Bellet

et al.[19] is not good enough for full 3D mapping using the high resolution detector

at high temperatures and there is no possibility to accurately determine the speci-men temperature by welding a thermocouple to it. There are also other furnaces available for high temperature synchrotron experiments, e.g., the furnace reported by Brokmeier et al.[20], but they are not suited for 3DXRD measurements due to space and other constraints.

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y

z

x

Sample

Beam stop

Area detector

Furnace

Incident x-ray beam

ω

η

2θ

Back side

F igur e 2.1: Schematic of the 3DXRD setup . S ho wn in the figur e ar e th e incident beam dir ection, the location of the specimen and the furnace ar ound it, and the 2D detector to recor d diffr action images. D uring the experiment, the specimen is rotated ar ound the v er tical axis o v er an angle ω in or der to bring all the gr ains into diffr action multiple times.

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§2.2 describes the challenges that arise from the specific requirements of the 3DXRD technique, while details of the furnace design are covered in §2.3. §2.4 covers the characteristics and performance of the furnace. An example of an experiment carried out using the furnace is presented in §2.5.

2.2 Design Requirements

The requirements of the furnace, which were kept in mind during its design and are in accordance to the 3DXRD technique, are listed as follows: Firstly, the fur-nace should permit rotation of the specimen around the vertical axis (ω) in order to be able to map all grains in the specimen and, at the same time allowing a high−resolution (e.g., Quantix [21]) detector to be placed at a small distance from the centre of the specimen (of the order of 8 mm). For the latter, the vertical dis-tance between the beam position and top of the furnace also needs to be kept low (of the order of 14 mm), due to space constraints dictated by the geometry of the high−resolution detectors. The X−ray absorption of the furnace chamber needs to be low enough in order not to interfere with the measurements. It should be possible to achieve high heating (up to 10◦_C_{/s) and cooling (up to 30}◦_C_{/s) rates} with accurate temperature measurement by welding at least one thermocouple to the metallic specimen. This is important to simulate in−situ heat treatments of metals and ceramics, for example, thermal cycles during welding in steels[22]. The temperature−time profile should be controlled accurately in order to simulate complex heat−treatments that are required for the production of e.g., modern TRansformation Induced Plasticity (TRIP) steels. Heating of the specimen (metallic or ceramic) should be uniform to reduce thermal gradients within the specimen (< 1◦_{C over the beam), needed to ensure that the measured temperature with} the thermocouple is the same as the temperature at the position of the synchro-tron beam. This is important because many metallurgical processes like diffusion of atoms depend exponentially on temperature. Environment control inside the furnace chamber should allow for working in a controlled atmosphere or under vacuum. Among others, this is important for steel research, because the carbon atoms can rapidly diffuse out of the steel in the presence of an oxygen−containing atmosphere, which would have a large effect on e.g. the transition temperature of the austenite−to−ferrite phase transformation in steel. The furnace should be able to heat the specimen up to 1500◦_{C while keeping the temperature in front of the} high−resolution detector lower than 40◦_{C to prevent damage to the detector. The} furnace should be able to operate at high temperatures (> 800◦_{C) continuously for}

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5 days, the typical duration of a 3DXRD beam time. The furnace design should also allow for thermal expansion of the specimen without deformation, while making sure that the specimen stands vertically straight. A deformation free specimen is very important to study, e.g., phase transformation kinetics in steels, where small deformations can lead to considerable changes in the nucleation and growth rates [23].

2.3 Furnace design

Although it was envisioned to design a furnace in which the beam could access the specimen over an angular range of 360◦_{, the required temperature of 1500}◦_C did not allow for a design with full rotation of the specimen. The furnace has been designed for a free rotation of the specimen over 90◦_{on the frontside and 90}◦_{on the} backside and the openings of the heating element dictate the maximum rotation angle. The asymmetric shape of the furnace results from the requirement of placing a high−resolution detector at a small distance from the specimen, while allowing for rotation of the specimen. The specimen is heated by radiation (from the heating coil) and convection (by the gas present in the specimen chamber). The schematic design and a real picture of the furnace are shown in Figure 2.2. The following subsections describe the design of the different parts in detail.

2.3.1 Furnace body

The furnace base and top are made of brass (Cu−64 wt%, Zn−36 wt%) to have a good thermal conductivity. The furnace weighs around 5 kg, excluding the weight of the vacuum assembly. The total height of the furnace is approximately 15 cm. The furnace is water cooled by the means of two channels, one in the base and one in the top. This assists in keeping the temperature variations of the furnace assembly low, thus providing good vertical stability of the specimen during experiments, which is important to ensure that the same specimen volume is measured during heating and cooling of the specimen. The thermal expansion of the specimen itself can be corrected for (see §2.3.3). Two vacuum feedthroughs are used to connect the thermocouples, one thermocouple through the base, on which the specimen is positioned and one thermocouple through the top, which is welded to the top of the specimen. The heating element is connected to a power supply also through the bottom feedthrough. A channel through the top can be used as an inlet for gas, regu-lated by a Bronkhorst digital EL−FLOW F−201CVTM_{electronic mass flow controller.} The furnace base has a channel connected to a pressure gauge, a vacuum pump and

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Figure 2.2: Schematics and a real image of the 3DXRD furnace. Different views are shown in

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x-ray beam direction top feedthrough for thermocouple supports for top inlet for gas

channel for cooling bottom feedthrough for thermocouple and power supply

channel for gas/vacuum

Figure 2.3: Cut−out view of the 3DXRD furnace. Difference parts of the furnace are indicated. One thermocouple is spot welded on the top of the specimen.

a gas outlet. The furnace base and the top are connected by two supports. Cut−out views of the furnace base, supports and top are shown in Figure 2.3. Two specially designed leader pins in each support assist in positioning the top at its designed place. In order to change the specimen, the furnace top can be removed with the help of two counter screws. A high precision translation table with micrometre screws is used to position the furnace in the centre of rotation. This is important for tomographic measurements from which the 3D structure of the specimen is reconstructed.

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x-ray beam direction

Kalrez o-rings

Kalrez o-ring Ceramic carrier Outer quartz tube Sample Heat shield Pt-Rh10% wire Furnace top Bottom thermocouple Inner quartz tube

7 mm

Figure 2.4: Cut−out view of the specimen chamber. The setup with two quartz tubes is shown. The X−ray beam direction and location of specimen is indicated. For changing the specimen, furnace top is removed and the specimen is dropped onto the bottom thermocouple. Geometry of the specimen can be seen with the step due to the change in diameter visible.

2.3.2 Specimen chamber

The specimen chamber, shown in Figure 2.4, consists of a region covered by a high purity (99.995%) quartz tube (1 mm thick to minimize background). To achieve optimum sealing of the specimen chamber, polymer O−rings are used between the quartz tube and the furnace base and top. Since the O−rings are less than 6 mm away from the heating element heated to temperatures greater than 1500 ◦_{C, the O}_{−rings also get heated to high temperatures. To ensure proper service,} Kalrez R _O_{−rings are used, which can withstand temperatures up to 317}◦_{C. Helium} present in the specimen chamber heats the specimen very effectively by convection.

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However, it also heats up the O−rings to high temperatures. Hence, in the case of long annealing times (> 5 hours) at high temperatures (> 800◦_{C), a different} setup has been designed. An additional quartz tube with two O−rings is placed on the inside of the original quartz tube. The O−rings for this quartz tube are even closer to the high temperatures and start to degrade quite soon. However, at the same time, they act as a barrier for helium access to the outer O−rings and thus keep the temperature of the outer O−rings low, resulting in longer vacuum stability. The construction with the double quartz window simultaneously acts as a heat shield to keep the temperature just outside the furnace low where an expensive high−resolution detector is positioned. The background signal during synchrotron radiation measurements from the quartz tubes is low for X−ray energies usually used for studies of metals, i.e. 50 keV or more.

2.3.2.1 Atmosphere control

Before heating up, the specimen chamber can be alternatingly purged with helium and kept under vacuum to get rid of any air remaining in the specimen chamber. The heat shield, made of pure tantalum or titanium, has two windows for the in-coming and transmitted/ diffracted beams and is put around the specimen and the heating coil in order to minimize heat losses to the surroundings. It simultaneously acts as a getter at high temperatures for any oxygen left behind. In addition, a small titanium ring is placed on the top of the heating element for the same purpose. The furnace can be operated under different specimen environments ranging from low pressure (up to 10−5 _{mbar) to controlled gases at desired pressure and also} in air. The sealing of the specimen chamber has been checked using a helium leak detector before, during and after a heat treatment (isothermal annealing at 1100◦_{C for 3 hours) to check the degradation of the O−rings. The rate of helium} leakage was found to be lower than 10−9_{mbar l}_{/s. No change in the leakage rate} was detected after the heat treatment, showing minimal or no damage to the outer O−rings.

2.3.3 Heating element and specimen holder

Although heating can be carried out by various methods such as induction heating, laser heating etc., the special requirements (a low thermal gradient and heating of a ceramic specimen etc.) of the furnace allowed for resistance heating only. In order to minimize the thermal gradients in the specimen, heating is done by passing DC current through a coil (Pt−10% Rh, φ = 0.25 mm) wound on a ceramic

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carrier (Al2O3tube, inner diameterφ = 2 mm) directly around the specimen (Figure 2.4). The Pt−Rh wire is chemically inert and can be heated to temperatures up to 1750◦_{C. The specimen is made out of a cylinder. The diameter changes three} times abruptly over the length of the specimen (see Figure 2.4). At the top and the bottom of the specimen, the diameter is the largest and just fits into the cylindrical ceramic carrier in order to position the specimen vertically straight and at the same time allow for free thermal expansion of the specimen in order to avoid any deformation of the specimen. At the location of the smallest diameter (φ = 1 mm), the 3DXRD measurements are performed. The intermediate diameter is used to serve as a reference point, which can be found with the synchrotron beam by monitoring the attenuation of the primary beam, which is higher in the thicker part of the specimen. Another method is to record the image of the specimen using the high−resolution detector. From the step edge, the specimen is moved with respect to the beam to the measurement location. In this way, the location where the synchrotron beam interacts with the specimen is exactly known. This gives the opportunity to perform ex−situ electron microscopy and EBSD measurements after the synchrotron measurements at exactly the same location where the synchrotron measurements were performed.

The specimen is lowered from the top on to the bottom thermocouple. S−type thermocouples (φ = 0.15 mm) are used for temperature measurements. This was chosen for two reasons: (i) S−type thermocouples are the most accurate at high temperatures and; (ii) the thin thermocouple wires help in minimizing the time lag for temperature readout and do not act as a heat sink. The inner diameter of the ceramic tube is chosen to allow for expansion of the specimen at higher temperatures. For minimizing heat loss to the surroundings, the volume of the specimen is kept small (8.5 mm3_{), which allows for a low power requirement and} also high heating and cooling rates. Small specimen volumes are sufficient for many investigations, because many of the microstructural changes in polycrystalline materials are in the order a few nanometres up to a millimetre.

2.3.4 Furnace control

Furnace control is achieved by using Eurotherm R _{3504 controller connected to a}

computer. Process modelling can be achieved on the controller or via the iTools R

program. For temperature control, either both or one of the two thermocouples at bottom and top of the specimen can be used. However, since the top thermocouple is welded directly on top of the specimen, it serves as the most accurate means

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of temperature measurement and control. The heat treatments and the gas flow can be programmed in the software. Real time recording of different parameters such as the temperature of the top and the bottom of the specimen, the gas flow conditions used, and the programmed parameters etc. can be carried out.

2.4 Furnace characteristics and performance

Due to the small size of the specimen, a very wide range of heating (up to 20◦_C_{/s) /} cooling rates can be achieved. Although quenching by helium can also be carried out, in most cases, the specimen can cool down at a rate of 30◦_C_{/s just by switching} off the power supply, with higher cooling rates possible by using a helium flow. Only 35 W of power is required to heat the specimen up to 1000◦_{C, with the limit for} maximum allowable power up to 80 W. The furnace can safely achieve temperatures up to 1500◦_{C. This has been tested by melting a steel specimen, which had a melting} point above 1500◦_{C. The use of a PID controller like Eurotherm} R _{3504 allows for}

accurate temperature control and stabilization. For the temperature constraint arising from the close proximity of the high−resolution detector with the specimen at 1000◦_{C, the temperature at a distance of 10 mm from centre of the specimen} reaches around 58◦_{C with only the outer quartz tube present. This is higher than} the allowable limit of 40◦_{C for the high}_{−resolution detector. This temperature} can be brought down to around 26◦_{C by using a cooling fan outside the furnace.} Furthermore, in order to reduce the heating by radiation, a reflective aluminium foil, 0.8µm in thickness, can be placed between the two quartz tubes. This is important when operating at temperatures above 1200◦_{C due to the increased} radiation. The temperature gradient in the specimen, measured during a thermal cycle of continuous heating from room temperature to 1000◦_{C in 2 min, holding} isothermally at 1000◦_{C for 10 min and continuous cooling to room temperature in} 5 min, was found to be less than 0.5◦_{C between top and middle of the specimen.}

2.5 Experiments

3DXRD experiments on the furnace were carried out in December 2008, July 2009, July 2010 and December 2010 at beamline ID11 at ESRF. Figure 2.5 shows the furnace installed at the beamline. The setup shown combines the use of both a high−resolution (Quantix [21]) and a medium−resolution (FReLoN [24]) detector simultaneously, in which the high−resolution detector is partially transparent. Quantix also has a slit to allow the primary X−ray beam through. Figures 2.6 (a, b)

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Figure 2.5: Photograph of the 3DXRD furnace installed at beam line ID11 at the ESRF. The

furnace is mounted on a rotation and translation table for positioning in the beam. The location of the high resolution (Quantix) and the medium resolution (FReLoN) detectors is indicated. The high resolution Quantix detector has a slit for the primary beam and is semi−transparent.

show two examples of diffraction images recorded simultaneously using the Quantix and the FReLoN detectors, respectively. The specimen (steel alloy of composition Fe−1 wt% Mn) was at 780◦_{C at the time of the measurement, with the Quantix a} dis-tance of 10 mm away from the centre of the specimen and the FReLoN a disdis-tance of 225 mm away. With cooling achieved using a fan, the measured temperature in front of the Quantix detector was 24◦_{C (2}◦_{C above ambient temperature). A beam size of} 1200×300 µm2_{was used. The X}_{−ray energy was calibrated to 78.395 keV using the} Laue−Laue setup and the specimen was rotated over an angle of 0.3◦_{during each} exposure. The exposure time for FReLoN and Quantix detectors was equal to 0.2 s and 5 s, respectively. Due to the close position of the Quantix detector with respect to the specimen, the position of the grains in the specimen strongly influence the position of the diffraction spots on the high−resolution detector, as shown in Figure

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Figure 2.6: Examples of diffraction images recorded from (a) high resolution Quantix

de-tector; (b) medium resolution FReLoN detector, simultaneously.

2.6 (a)[25]. In the case of the FReLoN detector, the position of the diffraction spots is influenced to a much smaller extent by the position of the diffracting grains in the specimen because the FReLoN detector is positioned at a distance of typically 250 mm away from the specimen. This is the reason why diffraction rings cannot be observed in Figure 2.6 (a), but can be observed in Figure 2.6 (b). More detailed treatment of the experiments with the FReLoN detector is given in Chapters 3 and 4.

Figure 2.7 shows a simulation of the thermal cycle during welding carried out on a TRIP steel specimen (composition, wt%: 0.19% C, 1.63% Mn, 1.1% Al and the rest Fe) and also examples of the diffraction images recorded using the FReLoN detector at the different stages during the heat treatment. The specimen was heated from the room temperature (Figure 2.7 A) at a rate of 15◦_C_{/s to 1000}◦_{C (Figure 2.7 B),} held isothermally at 1000◦_{C for 60 s (Figure 2.7 C and D) and quenched (Figure 2.7}

E) at an average rate of 15◦_C_{/s to room temperature (Figure 2.7 F). The maximum} cooling rate was 36◦_C_{/s from 1000}◦_{C to 600}◦_{C. At lower temperatures, the cooling} rate was reduced to simulate welding conditions[22]. Diffraction images were recorded every 0.3 seconds.

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0 200 400 600 800 1000 1200 06:43. 2 08:09. 6 09:36. 0 11:02. 4 12:28. 8 13:55. 2 15:21. 6 16:48. 0 18:14.4 90 180 270 360 450 540 630 720 0 200 400 600 800 1000 1200 _{e, °C} _Temperatur Time, s A B C D E F 0 A B C D E F F igur e 2.7: E xample of diffr action images recor ded during welding thermal cycle on a TRIP steel. A − diffr action image recor ded fr om the initial TRIP micr ostructur e. B − diffr action image recor ded during heating. C − diffr action image recor ded at the beginning of isothermal holding. D − diffr action image recor ded at the end of isothermal holding. E − diffr action image recor ded during quenching. F− diffr action image of the final micr ostructur e at room temper atur e. D ar ker regions repr esent pix els with positiv e intensity . P resence of both austenite and ferrite diffr action spots can be seen in the diffr action images.

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During the experiments, the stability of the setup was checked by carrying out various experiments on a single specimen for 32 hours. A specimen was kept at 900◦_{C for a total of 12 hours and was heated up and cooled down from and to 600} ◦_{C four times. At the end of the experiments, no oxidation was observed on the} specimens and the specimen chamber was still leak tight in each case. This shows that the furnace is ideal for long experiments without the need of changing the specimen while decreasing the beam time wasted for changing specimens.

2.6 Future developments

In future, the furnace can be modified to achieve a field of view of 360◦_{(except for a} shadow of the heating coil), by removing the ceramic carrier and the heating coil wound on it (see Figure 2.4 for details). These will be replaced by only a heating coil of Pt−10% Rh, the top and its supports will be removed and the furnace chamber will be covered with closed quartz. This setup would be very useful for techniques such as DCT, which require a full 360◦_{rotation around the specimen axis. However,} due to the reduced efficiency of heating from the wire, the maximum specimen temperature in this setup that can be reached would be limited to around 900◦_C due to absence of a ceramic carrier for heat transfer to the specimen.

In addition, the furnace chamber has been modified to fit the requirements for SAXS experiments. In the setup described here, while operating at low X−ray energies (< 20 keV), absorption and scattering signal from the quartz tubes can become significant. In such cases, the quartz tube is replaced by a specially de-signed steel enclosure with two slits for the incident beam and scattered signal. The two slits are covered with Mica foils (to ensure minimum absorption or scattering signal from the foils), held in place using a plate−on−plate design and sealed using high−temperature O−rings. However, the maximum specimen temperature in this setup is limited by the limited high−temperature stability of the Mica foils.

2.7 Conclusions

A furnace designed for carrying out synchrotron radiation experiments for polycrys-talline materials research is presented. The furnace combines various requirements of materials science research, such as fast heating (20◦_C_{/s) and cooling rates (30} ◦_C_{/s), high operating temperatures (1500}◦_{C), low temperature gradients across} the specimen (< 1◦_{C), accurate temperature control and control of the specimen}

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environment (vacuum up to 10−5 _{mbar or gas filled), with the requirements of} synchrotron radiation techniques (for example 3DXRD, SAXS and PCT), such as rotation of the specimen over a wide angular range (90◦_{), space and temperature} constraints from the detectors and low X−ray absorption of the furnace chamber. With the help of examples, the various strengths of the furnace are illustrated. A number of demonstrative examples of the furnace in use for metals research are shown in Chapters 5, 6 and 7. Its unique characteristics open up new fields of re-search in materials science such as studies of the kinetics of phase transformations and the simulation of welding conditions in metals. Although designed for metals research using the 3DXRD technique, the furnace is expected to prove to be an invaluable tool for ceramics research and be applied to other synchrotron radiation techniques as well.

2.8 Bibliography

[1] J. Baruchel, J. Y. Buffiere, P. Cloetens, M. Di Michiel, E. Ferrie, W. Ludwig, E. Maire, and L. Salvo, “Advances in synchrotron radiation microtomography,”

Scripta Materialia, vol. 55, no. 1, pp. 41–46, 2006.

[2] J. Baruchel, P. Bleuet, A. Bravin, P. Coan, E. Lima, A. Madsen, W. Ludwig, P. Pernot, and J. Susini, “Advances in synchrotron hard X-ray based imaging,”

Comptes Rendus Physique, vol. 9, no. 5-6, pp. 624–641, 2008.

[3] G. E. Ice and B. C. Larson, “3D X-ray crystal microscope,” Advanced

Engineer-ing Materials, vol. 2, no. 10, pp. 643–646, 2000.

[4] B. C. Larson, W. Yang, G. E. Ice, J. D. Budai, and J. Z. Tischler, “Three-dimensional X-ray structural microscopy with submicrometre resolution,”

Nature, vol. 415, no. 6874, pp. 887–890, 2002.

[5] D. J. Jensen, E. M. Lauridsen, L. Margulies, H. F. Poulsen, S. Schmidt, H. O. Sorensen, and G. B. M. Vaughan, “X-ray microscopy in four dimensions,”

Materials Today, vol. 9, no. 1-2, pp. 18–25, 2006.

[6] D. J. Jensen, S. E. Offerman, and J. Sietsma, “3DXRD characterization and modeling of solid-state transformation processes,” MRS Bulletin, vol. 33, no. 6, pp. 621–629, 2008.

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[7] S. E. Offerman, N. H. van Dijk, J. Sietsma, S. Grigull, E. M. Lauridsen, L. Mar-gulies, H. F. Poulsen, M. T. Rekveldt, and S. van der Zwaag, “Grain nucle-ation and growth during phase transformnucle-ations,” Science, vol. 298, no. 5595, pp. 1003–1005, 2002.

[8] E. M. Lauridsen, D. J. Jensen, H. F. Poulsen, and U. Lienert, “Kinetics of in-dividual grains during recrystallization,” Scripta Materialia, vol. 43, no. 6, pp. 561–566, 2000.

[9] A. King, G. Johnson, D. Engelberg, W. Ludwig, and J. Marrow, “Observations of intergranular stress corrosion cracking in a grain-mapped polycrystal,”

Science, vol. 321, no. 5887, pp. 382–385, 2008.

[10] W. Ludwig, S. Schmidt, E. M. Lauridsen, and H. F. Poulsen, “X-ray diffraction contrast tomography: A novel technique for three-dimensional grain mapping of polycrystals. I. Direct beam case,” Journal of Applied Crystallography, vol. 41, no. 2, pp. 302–309, 2008.

[11] G. Johnson, A. King, M. G. Honnicke, J. Marrow, and W. Ludwig, “X-ray dif-fraction contrast tomography: A novel technique for three-dimensional grain mapping of polycrystals. II. The combined case,” Journal of Applied

[12] S. Schmidt, S. F. Nielsen, C. Gundlach, L. Margulies, X. Huang, and D. J. Jensen, “Watching the growth of bulk grains during recrystallization of deformed

metals,” Science, vol. 305, no. 5681, pp. 229–232, 2004.

[13] L. Margulies, G. Winther, and H. F. Poulsen, “In-situ measurement of grain rota-tion during deformarota-tion of polycrystals,” Science, vol. 291, no. 5512, pp. 2392– 2394, 2001.

[14] J. V. Bernier, N. R. Barton, U. Lienert, and M. P. Miller, “Far-field high-energy diffraction microscopy: A tool for intergranular orientation and strain analysis,”

Journal of Strain Analysis for Engineering Design, vol. 46, no. 7, pp. 527–547,

2011.

[15] B. Jakobsen, H. F. Poulsen, U. Lienert, J. Almer, S. D. Shastri, H. O. Sorensen, C. Gundlach, and W. Pantleon, “Formation and subdivision of deformation structures during plastic deformation,” Science, vol. 312, no. 5775, pp. 889–892, 2006.

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[16] J. Oddershede, S. Schmidt, H. F. Poulsen, H. O. Srensen, J. Wright, and W. Re-imers, “Determining grain resolved stresses in polycrystalline materials us-ing three-dimensional X-ray diffraction,” Journal of Applied Crystallography, vol. 43, no. 3, pp. 539–549, 2010.

[17] A. D. Rollett, S. B. Lee, R. Campman, and G. S. Rohrer, “Three-dimensional characterization of microstructure by electron back-scatter diffraction,”

An-nual Review of Materials Research, vol. 37, pp. 627–658, 2007.

[18] L. Margulies, M. J. Kramer, R. W. McCallum, S. Kycia, D. R. Haeffner, J. C. Lang, and A. I. Goldman, “New high temperature furnace for structure refinement by powder diffraction in controlled atmospheres using synchrotron radiation,”

Review of Scientific Instruments, vol. 70, no. 9, pp. 3554–3561, 1999.

[19] D. Bellet, B. Gorges, A. Dallery, P. Bernard, E. Pereiro, and J. Baruchel, “A 1300 K furnace for in-situ X-ray microtomography,” Journal of Applied

[20] H. G. Brokmeier, S. B. Yi, N. J. Park, and J. Homeyer, “In-situ texture analysis using hard X-rays,” in 2nd International Conference on Texture and Anisotropy

of Polycrystals, vol. 105, pp. 55–60, 2005.

[21] A. Koch, C. Raven, P. Spanne, and A. Sinigrev, “X-ray imaging with submicro-meter resolution employing transparent luminiscent screens,” Journal of the

Optical Society of America A, vol. 15, pp. 1940–1951, 1998.

[22] M. Amirthalingam, M. Hermans, and I. Richardson, “Microstructural devel-opment during welding of silicon- and aluminum-based transformation-induced plasticity steels-inclusion and elemental partitioning analysis,”

Me-tallurgical and Materials Transactions A - Physical Metallurgy and Materials Science, vol. 40A, no. 4, pp. 901–909, 2009.

[23] H. Landheer, S. E. Offerman, R. H. Petrov, and L. A. I. Kestens, “The role of crystal misorientations during solid-state nucleation of ferrite in austenite,”

Acta Materialia, vol. 57, no. 5, pp. 1486–1496, 2009.

[24] J. C. Labiche, O. Mathon, S. Pascarelli, M. A. Newton, G. G. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The fast readout low noise cam-era as a versatile X-ray detector for time resolved dispersive extended X-ray absorption fine structure and diffraction studies of dynamic problems in

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materials science, chemistry, and catalysis,” Review of Scientific Instruments, vol. 78, p. 091301, 2007.

[25] H. F. Poulsen and S. Schmidt, “Reconstruction of grain boundaries in poly-crystals by filtered back-projection of diffraction spots,” Journal of Applied

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P

RE

−

PROCESSING OF

3DXRD

DIFFRACTION DATA

Abstract

A procedure for the pre−processing of data acquired using 3−Dimensional X−Ray Diffraction (3DXRD) is presented. The procedure deals with the pre−processing of the data for input in the algorithms presented in Chapter 4 for the determination of grain characteristics. An algorithm is presented for accurate identification of overlapping diffraction peaks from X−ray diffraction images, which has been a long−standing issue for experiments of this type. The algorithm works in two stages, namely, identification of overlapping peaks by using a seeded−watershed algorithm and then fitting the peaks with a pseudo−Voigt shape function to yield accurate centre−of−mass position and integrated intensity of the peaks. Regions consisting of up to six overlapping peaks can be successfully fit. Two simulations are used to verify the results of the algorithms. An example of the processing of diffraction images acquired in a 3DXRD experiment with a specimen consisting of more than 1600 grains is shown. Furthermore, a procedure for the determination of the parameters of the experimental setup (global parameters) without the need of a calibration specimen is described and validated using simulations. This is immensely beneficial for simplifying the experiments and the subsequent data analysis.

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3.1 Introduction

In the last decade, a number of techniques have been developed for non−destructive characterization of three−dimensional (3D) microstructures of polycrystalline ma-terials using high energy X−ray diffraction available at 3rd_{generation synchrotron} sources. A few examples include Differential Aperture X−ray Microscopy, DAXM [1], 3D X−Ray Diffraction microscopy, 3DXRD [2, 3], and Diffraction Contrast Tomo-graphy, DCT[4]. DAXM has nanometre resolution, but requires time−consuming acquisition strategies and is limited to relatively small volumes. 3DXRD and DCT, which can be considered a variant of 3DXRD, allow for the simultaneous character-ization of volume, crystallographic orientation and stresses in bulk microstructures with micrometre resolution. These two techniques have been used to investigate changes in polycrystalline materials at the microstructural level and have led to better understanding of the complex processes involved during heat treatments and deformation of materials[2, 5, 6, 7, 8]. In general, 3DXRD can be used in three modes. In the ‘fast−mode’, the average characteristics of each grain (that is, volume, centre−of−mass position, crystallographic orientation and average strain state) can be obtained with a time resolution of the order of a few minutes. In the ‘slow−mode’, grain shape, position and crystallographic orientation can be mapped with a time resolution of the order of a few hours. In the ‘high−resolution mode’, crystallographic orientation of and strains within individual crystals can be mapped in reciprocal space, again with a time resolution of the order of a few hours. A full overview of the technique can be found elsewhere[9].

The present chapter deals with the analysis of data acquired using ‘fast−mode’ 3DXRD. In this technique, if the number of grains in the volume illuminated by the X−ray beam is limited, the number of grains that satisfy the Bragg diffraction condition for an orientation of the specimen is also limited, leading to individual diffraction spots from individual grains, which can then be used to determine the characteristics of the grains. Thus, the technique is intermediate between single crystal diffraction and powder diffraction. As the number of grains in the illuminated volume increases (or in the case of textured/deformed materials), diffraction spots from different grains overlap. This problem has been highlighted as a long−standing limitation of the 3DXRD type of experiments a number of times in References[9, 10, 11, 12, 13, 14]. A strategy widely used to overcome this problem has been to reduce the illuminated volume and scan multiple volumes (so−called ‘layers’) in the specimen, see, e.g, Oddershede et al. [11]. However, this results in worse time resolution during the acquisition of data as compared to a

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single layer. In the present chapter, a data analysis strategy, using which, data acquired from a large number of grains (of the order of thousands) and consisting of peak overlap can be successfully analysed with low computation requirements, is presented. This should lead to a substantial improvement in acquisition times, allowing, in turn, for tracking of thousands of grains simultaneously during in−situ measurements with improved time resolution.

3.2 3DXRD technique

The setup of the 3DXRD technique operating in ‘fast−mode’ is schematically shown in Figure 3.1 (a). A collimated and monochromatic beam with a rectangular cross−section1 _{is incident on the specimen. The specimen, of a known} struc-ture, is placed on the top of a rotation setup comprising of anω rotation stage (ω is the rotation around an axis perpendicular to the X−ray beam) and multiple

x, y and z translation tables for alignment. The specimen can also be placed in a

special environment for carrying out in−situ studies (e.g. the multipurpose furnace presented in Chapter 2). In any orientation of the specimen, the grains that satisfy the Bragg diffraction condition will generate a diffracted beam leading to a signal on the two−dimensional detector. In order to get all the grains in the illuminated volume under diffraction a number of times, the specimen is rotated around the

ω−axis. Since real materials consist of imperfect grains having variations in the

orientation (so−called mosaicity), the specimen is rotated by an angle ∆ω during each exposure to bring all parts of the grains in diffraction.2

In the ‘fast−mode’ 3DXRD, diffraction images are acquired by using only the far−field detectors with a pixel size of the order of 50 µm placed relatively far from the specimen (∼20−50 cm), due to which the diffraction spots appear at the in-tersection of the Debye−Scherrer cones (with opening angle 4θ ) and the detector plane, so called diffraction rings. In case the mosaicity of a grain is sufficiently large, or∆ω sufficiently low (described in §3.3.3), diffraction spots are observed in multiple diffraction images, with spots in each image being characterized as

dif-fraction peaks. For un−deformed grains (or grains with homogeneous intra−grain

1_{The shape of the beam is usually of the following form: (a) pencil (both horizontal and vertical}

dimensions of the beam are smaller than the average grain size), (b) slice (either horizontal or vertical dimension of the beam is increased to illuminate multiple grains only in 1D) and, (c) rectangular (both vertical and horizontal dimensions of the beam are larger than the average grain size to fully illuminate multiple grains).

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y z x Sample Beam stop Area detector Furnace

Incident x-ray beam _ω

η 2θ γ η O A B β Tilted detector Ideal detector Sample Diffraction ring on tilted detector Diffraction ring on ideal detector (b) (c) Sample 2θ O A B R R’ Lsd x y z δ δ η’ z’ (a) y’ Back side

Figure 3.1: Overview of the 3DXRD technique. (a) Schematic of the ‘fast−mode’ 3DXRD setup with the far−field detector. The dimensions are not to scale. The angles 2θ , ω and η are defined. (b, c) Definition of detector tilts. (b) Schematic view of the diffraction ring as observed on the tilted detector and as it should appear on a perfect detector. (c) 2D view of the schematic shown in (b).

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deformations), these diffraction peaks can be well approximated by a pseudo−Voigt function. Depending on the relative position of the diffracting grains in the speci-men with respect to the centre of rotation of the specispeci-men, the diffraction peaks are displaced relative to the ideal position on the diffraction rings. From the dif-fraction spots, the following characteristics of grains can be derived: (i) volume of the grains by using the integrated intensity of the diffraction spots; (ii) crystallo-graphic orientation of the grains by using the position of the diffraction spots on the diffraction ring and inω; (iii) position of the centre−of−mass of the grains by using the position of the diffraction spots with respect to the idealized position on the diffraction ring and; (iv) strain state of the grains by using the position of the diffraction spots on the detector and inω. The derivation of these characteristics is dealt with in Chapter 4.

In order to calculate the above−mentioned grain characteristics, diffraction images need to be pre−processed to accurately determine the centre−of−mass position (on the detector and inω) and the integrated intensity of diffraction spots and the global (independent of grain) parameters (wavelength of the X−ray beam, lattice parameter of the specimen, specimen−to−detector distance, wedge3_{of the} incoming beam with respect to the axis of rotation, tilts of the detector and centre of the beam on the detector) of the experiment need to be determined. Determination of the spot characteristics is described in §3.3 and fitting of the global parameters by using the position of the diffraction spots on the detector and inω is explained in detail in §3.4.

3.3 Characterization of peaks

In the case of diffraction images containing no overlap, a direct approach can be adopted. A single thresholding operation on the corrected diffraction images yields the position of all connected pixels in an image, each set of connected pixels being a diffraction peak. Various characteristics of the diffraction peaks such as integrated intensity, centre−of−mass position etc. can then be calculated [11, 13]. For non−overlapping peaks, Edmiston et al. [13] use a 3D Gaussian function for fitting the peak shape after coordinatizing the peak position in (2θ , η, ω) from (y, z,

ω). However, if overlapping peaks are present, this procedure leads to erroneous

results because each set of overlapping peaks is counted as one and thus the calculated centre−of−mass position and the integrated intensity of the peaks are

3_{Wedge is defined as the deviation of the axis of rotation from the ideal rotation axis orthogonal}

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wrong. This can be avoided by combining the results of multiple thresholds,4_which can resolve the overlapping peaks, but the integrated intensity of the peaks is still wrong, and in the cases with strong overlap, so is the centre−of−mass position of the peaks. In order to overcome such problems, the shape of the diffraction peaks can be fit using an ideal shape (in 2D or 3D) to obtain refined characteristics of the diffraction peaks. To this end, the program DIGIgrain[15] employs a 3D connectivity peak search to characterize overlapping diffraction peaks. In this section, a complementary technique is presented, which employs a 2D peak fitting routine in order to efficiently resolve and characterize overlapping peaks.

The diffraction images acquired in 3DXRD usually consist of a large number of pixels with background intensity interspersed with a considerably lesser number of pixels with signal (intensity higher than the background) from the diffracting grains on the diffraction rings. For diffraction data with peak overlap, the determination of the peak characteristics by fitting the diffraction image as a whole with multiple diffraction peaks can be computationally very intensive and un−wise because: (i) a significant number of pixels (background) are not of interest and thus need not be processed; (ii) the parameters (width, height etc.) defining the shape of each peak to be used during fitting can vary significantly depending on the size of the diffracting grains, orientation spread within the grains and the position of the diffraction spots and; (iii) depending on the number of peaks in the diffraction image, the number of parameters to fit can be very large. In the methodology presented here, these issues are avoided as follows: (i) the background pixels in the image are removed by applying a threshold to the diffraction image; (ii) each set of interconnected pixels with intensity above the threshold is considered separately during fitting as a set of overlapping peaks (a considerably smaller set compared to all the peaks present in the diffraction image) and; (iii) guesses for the parameters to be fit are obtained by using the information about the connected pixels. The procedure for the determination of the characteristics of the peaks is then divided into three parts:

(i) A peak searching algorithm to carry out the thresholding operation, to de-termine the number of peaks in each set of connected pixels using the number of maxima and to determine approximate characteristics of the peaks (integrated intensity and centre−of−mass position) using a seeded−watershed operation [16].

4_{The algorithm for peak searching using multiple thresholds can be provided upon request.}

Alternatively, an implementation of peak searching using multiple thresholds is available in the

Fable package athttp://fable.svn.sourceforge.net/viewvc/fable/ImageD11/trunk/